UK banks lost a test case a few years ago that led to a lot of people getting back however many years of overdraft charges, plus interest. The same thing happened a bit later with “payment protection insurance”, intended to cover loan repayments if you lost your job, but with so many exclusions as to be almost worthless.
The end result was something like a forced savings policy. Cue people who avoided the initial trap wondering where their free money is.
You have to wonder sometimes.
The problem with that is that the same argument goes through in exactly the same way with any stronger system replacing PA. You might first try something like adding a rule “if PA proves that PA proves S, then S”. This solves your original problem, but introduces new ones: there are now new statements that your system can prove that it can prove, but that it can’t prove. Eliezer discusses this system, under the name PA+1, in You Provably Can’t Trust Yourself .